In principle, we should be able to find all the QNMs by finding the roots to (REF ), but in practice our numerical methods only find a few roots using this single equation. For this reason it is useful to “invert" the equation.
The \(n\) -th inversion, for \(n=2, 3, \dots \) , of this continued fraction is defined to be [1]} :
\(\beta _n^{(4)} - \frac{\alpha _{n-1}^{(4)} \gamma _n^{(4)}}{\beta _{n-1}^{(4)} -} \frac{\alpha _{n-2}^{(4)} \gamma _{n-1}^{(4)}}{\beta _{n-2}^{(4)} -}\cdots \frac{\alpha _1^{(4)}\gamma _2^{(4)}}{\beta _1^{(4)}}=\frac{\alpha _{n}^{(4)} \gamma _{n+1}^{(4)}}{\beta _{n+1}^{(4)} -} \frac{\alpha _{n+1}^{(4)} \gamma _{n+2}^{(4)}}{\beta _{n+2}^{(4)} -} \frac{\alpha _{n+2}^{(4)}\gamma _{n+3}^{(4)}}{\beta _{n+3}^{(4)}-}\cdots \) 
